Timaeus
Yourgrau's book "Disappearance of Time," notes and excerpts:
Firstly, on Goedel's Incompleteness theorem: Hilbert's math assumed a fundamental equivalence of Form and Content, so that all Content, all meaning (semantics) could be consistently and Completely represented as pure structure, pure Form, pure Syntax. G's breakthrough was to use Form to elucidate Form's limit in expressing Content. G's theorem, then, is a Formalization of the dialectic of Form and Content, and the inability of the latter to complete reduce to the former. G's complaint of the "formalists" was that they "considered formal demonstrability to be an analysis of the concept of mathematical truth." Instead, formalisms cannot "speak for themselves" about their own significance; they need something more to be complete, something informal or "intuitive."
"Formal questions asks us to supply content of the answer, whereas informal questions demand that we supply both form and content." G's genius was converting the informal into the formal, then introducing constructions that limit the allowable "contentual interpretations" of the Form's domain.
Because of this, one must make a sharp distinction between epistemology and ontology, because the latter cannot be reduced to the former. This is the essence of G's philosophy: the distinction between proof and truth.
G, along with Plato, regarded the world of time as "ontologically suspect," and his philosophical strivings were efforts to merge the realms of the eternal and temporal.
For Kant, the proper image of time was a line generating itself: "by this mode of depicting it alone could we know the singleness of its dimension." For Aristotle this dynamic model of time leaves the past "fixed and continuously supplemented" and the future at least partially indeterminate; the place where they meet, the point of Chiaroscuro, is the Now. But these ideas of direction, movement, trajectory, of flux and points and instants -- these ideas are at the least metaphorically geometric, and at most literally geometric. This "geometrization of physics" reached an apotheosis with Einstein's theory of relativity, which represents a complete "mathematization of time" (geometrization of the temporal). Thereafter, the concept of time was completely absorbed -- completely reduced -- to structure, geometry, Form.
The spatialization of time has a serious implication: if time is susceptible to a purely geometric treatment, then all coordinates of temporal universe are ontologically neutral. Ontological-neutrality is the essence of "space", the essence of extension; therefore, if Einstein was right, an event's location in time has no effect whatsoever on the status of that event's existence. To spatialize time is to collapse the temporal mode of being -- the mode of becoming, of actualization out of the potential -- into platonic objects that, instead of coming into being successively, exist eternally.
But as we've seen, as G showed us, one cannot replace content with form. And this was the meaning of G's solution to E's relativity theory. In his solution G proved the possibility of nonstandard "unlucky universes", the so-called rotating or R-universes, where "objective lapse of time is an illusion" ( i.e. all world-lines loop back to connect to themselves, and time-travel through space is possible) and "even though it is an empirical question whether or not ours is an R-universe (it's not), the collapse of the objectivity of genuine temporal succession in a universe differing from ours only in certain non-lawlike features concerning the cosmic distribution of matter and motion shows that in our universe, too, time is merely 'ideal.' "
So G illuminated an inconsistency. Wittgenstein once said, "The world is all that is the case. It is the totality of facts, not of things." But if time and change are real, Reality grows by accretion of facts, and this would mean that the position in time does carry ontological weight. Tensed existence "entails that we can, at different times, rightly assert about the same thing that it exists and that it does not exist." ("Reality consists of an infinity of layers of "the now" which come into existence successively.") "What is temporally, both is and is not." By proving the possibility of R-universes within Einstein's theoretical domain, G was not trying to prove the non-existence of "objective becoming." Instead, he was demonstrating that Einstein's geometric formalism of time could not possibly be contentually complete: time really is "branchwise asymmetric", time-lapse really is objectively true, and E's theory disallowed it. G believes that time is not a mere illusion, and since E's construction of relativity admitted universes where time-lapse was "provably meaningless", there must be more to time than geometry.
The "direction of time" cannot be captured in purely spatio-geometrical terms. In space motion is relative and reversible. In time movement is unidirectional and irreversible. Events cannot be de-actualized or re-potentialized.
additional notes:
This full Meaning of time is better heard than seen. One needs a different explanation for the "unity of melody, successive in time, defined by time" from the "nonsuccessive unity of chords played spatially." One can grasp a melody structurally (tenselessly) by studying the score, but a piece of music must be performed to be completely comprehended: music is essentially a temporal entity. Music is not music unless it is being performed in real time.
"Eternity for Plato is not temporal duration; it is not a measure but a mode of being." Eternity is not "everlastingness" but "genuineness." The fundamental analogy for time is "a moving image of eternity." So temporal being is not complete being.
Time separates the potential qua potential from its realization qua actual, expressed in the form of motion. Being eternal means being motionless. [me: but even 2+2=4 is not eternal. in the empty set, in the Nothing, even that ceases to exist (though such truths will be the last to decay.]
There are two types of truths. There is the A-series truth, which is constantly in flux due to the nunc fluens -- the flowing now -- and the B-series truths, in which a truth about an event is fixed. A B-theoretic truth is "Kennedy was shot in 1963". An A-theoretic truth is "It is now 2007." The former's truth is fixed. The latter's truth is unable to be fixed permanently. For time to be real, both truth-series have to exist -- have to have ontological weight.
Random thought: memory multiplies the dimensionality of the Past's causality; without memory, causation is temporally linear only. With memory, the past can communicate -- and affect -- the present multi-dimensionally. If the present is the coded past, memory expands and complicates the code.
Firstly, on Goedel's Incompleteness theorem: Hilbert's math assumed a fundamental equivalence of Form and Content, so that all Content, all meaning (semantics) could be consistently and Completely represented as pure structure, pure Form, pure Syntax. G's breakthrough was to use Form to elucidate Form's limit in expressing Content. G's theorem, then, is a Formalization of the dialectic of Form and Content, and the inability of the latter to complete reduce to the former. G's complaint of the "formalists" was that they "considered formal demonstrability to be an analysis of the concept of mathematical truth." Instead, formalisms cannot "speak for themselves" about their own significance; they need something more to be complete, something informal or "intuitive."
"Formal questions asks us to supply content of the answer, whereas informal questions demand that we supply both form and content." G's genius was converting the informal into the formal, then introducing constructions that limit the allowable "contentual interpretations" of the Form's domain.
Because of this, one must make a sharp distinction between epistemology and ontology, because the latter cannot be reduced to the former. This is the essence of G's philosophy: the distinction between proof and truth.
G, along with Plato, regarded the world of time as "ontologically suspect," and his philosophical strivings were efforts to merge the realms of the eternal and temporal.
For Kant, the proper image of time was a line generating itself: "by this mode of depicting it alone could we know the singleness of its dimension." For Aristotle this dynamic model of time leaves the past "fixed and continuously supplemented" and the future at least partially indeterminate; the place where they meet, the point of Chiaroscuro, is the Now. But these ideas of direction, movement, trajectory, of flux and points and instants -- these ideas are at the least metaphorically geometric, and at most literally geometric. This "geometrization of physics" reached an apotheosis with Einstein's theory of relativity, which represents a complete "mathematization of time" (geometrization of the temporal). Thereafter, the concept of time was completely absorbed -- completely reduced -- to structure, geometry, Form.
The spatialization of time has a serious implication: if time is susceptible to a purely geometric treatment, then all coordinates of temporal universe are ontologically neutral. Ontological-neutrality is the essence of "space", the essence of extension; therefore, if Einstein was right, an event's location in time has no effect whatsoever on the status of that event's existence. To spatialize time is to collapse the temporal mode of being -- the mode of becoming, of actualization out of the potential -- into platonic objects that, instead of coming into being successively, exist eternally.
But as we've seen, as G showed us, one cannot replace content with form. And this was the meaning of G's solution to E's relativity theory. In his solution G proved the possibility of nonstandard "unlucky universes", the so-called rotating or R-universes, where "objective lapse of time is an illusion" ( i.e. all world-lines loop back to connect to themselves, and time-travel through space is possible) and "even though it is an empirical question whether or not ours is an R-universe (it's not), the collapse of the objectivity of genuine temporal succession in a universe differing from ours only in certain non-lawlike features concerning the cosmic distribution of matter and motion shows that in our universe, too, time is merely 'ideal.' "
So G illuminated an inconsistency. Wittgenstein once said, "The world is all that is the case. It is the totality of facts, not of things." But if time and change are real, Reality grows by accretion of facts, and this would mean that the position in time does carry ontological weight. Tensed existence "entails that we can, at different times, rightly assert about the same thing that it exists and that it does not exist." ("Reality consists of an infinity of layers of "the now" which come into existence successively.") "What is temporally, both is and is not." By proving the possibility of R-universes within Einstein's theoretical domain, G was not trying to prove the non-existence of "objective becoming." Instead, he was demonstrating that Einstein's geometric formalism of time could not possibly be contentually complete: time really is "branchwise asymmetric", time-lapse really is objectively true, and E's theory disallowed it. G believes that time is not a mere illusion, and since E's construction of relativity admitted universes where time-lapse was "provably meaningless", there must be more to time than geometry.
The "direction of time" cannot be captured in purely spatio-geometrical terms. In space motion is relative and reversible. In time movement is unidirectional and irreversible. Events cannot be de-actualized or re-potentialized.
additional notes:
This full Meaning of time is better heard than seen. One needs a different explanation for the "unity of melody, successive in time, defined by time" from the "nonsuccessive unity of chords played spatially." One can grasp a melody structurally (tenselessly) by studying the score, but a piece of music must be performed to be completely comprehended: music is essentially a temporal entity. Music is not music unless it is being performed in real time.
"Eternity for Plato is not temporal duration; it is not a measure but a mode of being." Eternity is not "everlastingness" but "genuineness." The fundamental analogy for time is "a moving image of eternity." So temporal being is not complete being.
Time separates the potential qua potential from its realization qua actual, expressed in the form of motion. Being eternal means being motionless. [me: but even 2+2=4 is not eternal. in the empty set, in the Nothing, even that ceases to exist (though such truths will be the last to decay.]
There are two types of truths. There is the A-series truth, which is constantly in flux due to the nunc fluens -- the flowing now -- and the B-series truths, in which a truth about an event is fixed. A B-theoretic truth is "Kennedy was shot in 1963". An A-theoretic truth is "It is now 2007." The former's truth is fixed. The latter's truth is unable to be fixed permanently. For time to be real, both truth-series have to exist -- have to have ontological weight.
Random thought: memory multiplies the dimensionality of the Past's causality; without memory, causation is temporally linear only. With memory, the past can communicate -- and affect -- the present multi-dimensionally. If the present is the coded past, memory expands and complicates the code.
posted by John Aristides at 3/18/2007 09:47:00 PM
1 Comments:
Lumsden and Wilson's 25th Anniversary Edition of "Genes, Mind, and Culture" has a graph illustrating my random thought on memory and dimensionality, which gives me faith in my intuition, but, I must admit, makes me less confident in my originality. It's on page 359.
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